12 Answers
12

Sensors are rectangular by tradition, based on the historically traditional shape of image media.

But there is a technology/business decision that drives them to be rectangular, also. Sensors are rectangular because they are made using semiconductor fabrication techniques. These techniques call for “printing” multiple sensor circuits onto a silicon wafer. Today these wafers can be 300 mm in diameter and manufacturers are moving toward 450 mm diameter (see here). A lot of sensors can be printed on wafers that large.

Sensors are tiled onto the wafer to efficiently use the space available and in a way that makes them easy to cut apart into “dies” (or the individual sensors, in this case). The process is called dicing. The most cost effective shape for dies is rectangular. Usually a saw or scribe is used to cut the wafers in straight lines. Imagine if the dies (sensors in this case) were supposed to be round (a wasteful and costly use of the material) or hexagonal (efficient use of the material but the cuts are not straight across the whole wafer). (See here for more info.)

B) Lenses made of high quality glass are generally ground using lathes. (This can be seen in this video. Watch around the 7:00 minute mark in particular. Sorry, it's in Japanese, but the video is very fascinating and revealing.) It is easier to spin, grind, and polish a round lens in these machines because there are no edges to catch on the tooling as the lens spins around. It also is consistent with the optical symmetry they are trying to achieve in the finished lens.

Lenses that are not round would generally be cut from round lenses, a step that adds cost to the production of the lens assembly. Lenses don’t need to be round. For heaven’s sake, most eyeglasses are not round! When your eyeglasses are made, you must be aware that the lens maker isn’t stocking a lens for every shape of eyeglass frame. He’s cutting or grinding round lenses to fit the frame.

Once the lens manufacturer has his round lenses, what would motivate him to cut it into a different shape? As many people have pointed out in various forums, the lens shape does not determine the image shape or quality (apart from diffraction caused by edges, which can be mitigated, and some second order aberration effects, maybe), and for the most part, every point on the lens can gather light from every point on the object and focus each point on the image plane. I’ve already pointed out that changing the shape of the lens adds cost. There really isn’t any practical reason (generally) for changing the shape.

Cylindrical is also the most convenient shape to make a lens barrel in terms of strength, and the ability to a helicoid for precise movement of the focussing groups. You could if you wanted to cut each lens to a rectangle, make a rectangular barrel with a rectangular aperture, but all you'd end up with is a very expensive lens that produces weird bokeh.
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Matt GrumJul 18 '13 at 9:41

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The sensors for very large optical telescopes are round (to not waste any of the very expensive image circle) but they're made by stitching together lots of small rectangular wafers.
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Matt GrumJul 18 '13 at 9:43

Last point, the RH-1 anamorphic lens was an april fools story - no such lens exists. But there are plenty of square front anamorphics made by Lomo.
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Matt GrumJul 18 '13 at 9:47

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Just an additional cool fact to a really nice answer: Lenses are kind of 'holographic'. If you take a lens and break it in half, you still get the whole image, just dimmer. You can take a lens shard, and you still get the whole image, just dimmer. You can put a lot of lens shards together and get a fresnel lens.
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Kaushik GhoseOct 4 '13 at 18:05

From the side of the manufacturer, it is easier and cheaper to manufacture spherical lens and easier to calibrate when you combined different lens to achieve a unique feature, e.g. macro, telephoto etc...

For the general users, most of us will definitely agree to say that it is more convenient to rotate circular lens than rectangular. Inside camera lenses, especially zoom lenses, some elements must adjust mostly by rotating (cheaper lenses) as you focus or zoom them. Rotating a non-circular lens is going to be tricky if you are also trying to control the orientation of the aberrations and diffraction spikes at the same time.

Trying to curve something flat is harder than making a curve of something round.

For the wide angle lenses, it has spherical shape to give better and wider perspective.

To focus on light with varying distance, it requires a circular lens as all points of light need to be focused on the same general area.

To produce images achieving maximum resolution (sharpness) the lens surface must be accurate to very high precision for the lens to deliver full resolution - small fractions of a wavelength of light. The grinding and polishing processes are only assured of producing lenses of the desired accuracy for circular lenses; it is extremely difficult though not impossible to achieve this accuracy for other shapes.

The most desirable properties of a lens are its ability to form sharp images without artifacts, and light gathering power especially in dim lighting. Both of these properties are maximized by circular lenses; only someone absolutely ignorant of optics theory would attempt to design any other shape.

Just one beef: lens elements never "have to" rotate (unless they're deliberately introducing astigmatism). It can be cheaper to let them rotate, but I've never owned a lens that didn't use captive followers on a straight path in addition to a positioning cam for moving elements.
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user2719Jul 15 '13 at 5:31

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Of course reasons 1, 3 and 6 wouldn't keep a manufacturer from producing a round lens and then sawing off the parts that focus light that doesn't hit the sensor anyway, say to save weight. For a long prime really only reasons 5 and 7 remain and maybe they could be explained in a little more detail (more than "it's just the way it is because ... optics").
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ChristianJul 15 '13 at 11:22

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Yeah, is like to hear more about the optics — assume we are "ignorant of optical theory", since that's the essential question here.
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mattdmJul 15 '13 at 11:43

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I guess another thing would be that to ensure a rectangular lens works would require an exact alignment with the sensor. Not required in the case of a circular lens since only the center needs to match, not the rotation
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AkashJul 15 '13 at 12:18

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This is a complete non-answer. Almost every single point is either meaningless or does not actually explain anything. What does 3 mean? Are 1 and 4 based on a confusion between round lenses and spherical optics? Why is it easier and cheaper to make spherical lenses? Why are the 'grinding and polishing processes only assured ... for circular lenses? What does it mean to say that round lenses 'give better and wider perspective'? This seems like a reasonable answer at first, but as soon as you read the words it's clear that it's just verbiage designed to 'sound right'.
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jwgJul 16 '13 at 13:22

One more reason: The light gathering capability is largely governed by the area, whereas some of the optical quality goes down (or it is more expensive to correct to the same level) with the maximum dimension. A circle minimizes the maximum dimension for a particular area.

Despite that, manufacturing concerns are the overriding reason. Fortunately, a circular lens is what you want for other reasons anyway.

A funny point is that the shape of the aperture (thus of the lens) affects the apparent shape of an out-of-focus light source (often called "bokeh"). You can see that looking at the custom bokeh images (http://www.wikihow.com/Make-a-Custom-Bokeh).

But, just as one can have a square or star-shaped aperture in a round lens, one could have a square lens with a round aperture.
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mattdmJul 15 '13 at 11:45

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@mattdm Then the square lens would need to be large enough to include the circle of the current lens inside the square - so now you are making a larger, heavier, more expensive (in terms of raw materials & manufacturing processes) lens. If not, the shape of the lens would act as a secondary aperture, just as with some lenses the aperture blades are not in the light path at all when the lens is wide open.
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Michael ClarkJul 15 '13 at 12:09

a benefit with those rectangular objectives would be that they need only to be 4 bladed, even the L lenses, and the Carl Zeiss ones. flo: why are those bracelets so darn expensive! I could consider one to match my lens coffee cup but it is silly expensive...
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Michael NielsenJul 15 '13 at 22:46

there are several more esoteric types of lenses, (lenslet arrays, kinoform lenses, etc..) But what is important to remember is that a lens is used to bend light, and there are many ways to do that using "diffractive" optics or tranditional glass like material. The reason for the design is usually functionality and production cost.

Let's say you use a rectangular lens rather than a cylindrical one. First off, the shape of the lens won't matter at all unless you have the aperture all the way open; on any slower setting, the approximately circular shape of the diaphram will be the determining factor. Assuming that you do have the aperture all the way open, the main effect will be as follows. You will have a certain depth of field. If object point A is at the correct distance to produce a pointlike image, then this point is still a point regardless of the rectangular shape of the lens. However, if object point B is at some other distance, we get a blur as the image of that point. The blur occurs because there is a bundle of light rays, and the bundle has some finite size where it intersects the film or chip. Since the lens is rectangular, this bundle is pyramidal, and the blur will be a rectangular blur rather than the usual circular one. For example, say you're photographing someone's face with a starry sky in the background. You focus on the face. The stars will appear as little fuzzy rectangles.

At very high magnifications (maybe with a very long lens that's effectively a small telescope), it's possible that you would also see diffraction patterns. In the example of the face with the starry background, suppose that we change the focus to infinity, putting the face out of focus. Wave optics would now predict that (in the absence of aberrations), the diffraction pattern for a star would be a central (order 0) fringe surrounded by a ring (first-order fringe) if you used a circular aperture, but a rectangular aperture would give a different pattern (more like a rectangular grid of fringes). In practice, I don't think a camera would ever be diffraction-limited with the aperture all the way open. Diffraction decreases as the aperture gets wider, while ray-optical aberrations increase, so aberration would dominate diffraction under these conditions.

@MichaelNielsen: Do you mean that it will only be approximately circular?
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Ben CrowellJul 15 '13 at 20:01

yes, thats what I mean. Normally you want to go for wide open to get perfect circles, which you then only get if you have a really good lens with no spherical aberrations.
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Michael NielsenJul 15 '13 at 20:02

Lenses were always being produced as rounded because it fits the manufacturing process the best. Making them square would involve at least very precise cutting afterwards, so that would make them much more expensive. (However square lenses are being produced for some special purposes)

You could ask why the sensor is square rather then round?

The answer to that is that our screens, film and in the end our photo paper is square in shape. We do not need round sensor if we need square photos!

Lenses don't have to be round. Look at the variety of shapes in which eyeglass frames appear.

However, all those lenses are a section cut from a stock lens which has spherical surfaces (ignoring, for a moment, lenses which correct for astigmatism).

And there is basically the answer. Any sort of assymetry would give your camera astigmatism: the inability to bring a point into focus vertically and horizontally at the same time.

A lens must provide a consistent focus along any rotational axis. If two parallel beams of light that are one cm apart horizontally hit the lens, they must focus at the same distance as two parallel beams which are one cm apart vertically.

to apply the same 'operation' to light coming in at any orientation you need a circularly symmetrical shape in order that you don't distort the spacial ratios between different points on the incoming image

lenses generally aim to concentrate light landing on their surface towards a single point. That point is 'slightly behind' the CMOS sensor in a camera, but its the same principle, and physics dictates that a lense cross-section shape achieves that. When you repeat it in all orientations you rotate around and get a flat-dome shape, like a lense

It's a similar reason to why satellite dishes are domed and not box-shaped

but it's definitely not because it's easier to manufacture. Raindrops and glass-balls have a lense-effect. Windows don't. Cubes and boxes of refractive material just don't have that effect.

Aside from creating really odd Bokkeh, a rectangular lens would also worsen the lens's vignetting and create asymmetrical resolution over the image area amongst other negative optical aberration effects. The light that is hitting any particular point on the sensor has come from a wide swath of glass - the light that's hitting a corner of the sensor didn't travel exclusively through the corresponding corner area of the lens elements on its way to the sensor (unless you consistently choose such small aperture that diffraction itself is substantially degrading image quality). Lens manufacturers go to great lengths to ensure that everything is symmetrical for the purpose image quality, even including the diaphragm. Low quality lenses may have a few aperture blades with flat edges making a very angular pentagon or hexagon iris...this can have a measurably negative effect on a lens's MTF chart (a measure of a lenses resolving capability) even in the center of the image. Move to better quality lenses and you'll find much more symmetrically round diaphragm opening...the apertures on those higher end multi-thousand dollar lenses that Canon & Nikon put out have very round diaphragms - that's just the aperture...do that to the glass and you'll be degrading the image that much more. The truly high end ($ 5-digit) lenses in cinematography have circular. Lens elements. This is all for the image quality throughout whole image area from center to corner. Regardless of whether the sensor is square, rectangular, round, or even star or crescent shaped, the lens - at least a really good one - would continue to be symmetrical (aka circular). Yes, manufacturing ease/cost and structural concerns would be adversely affected and complicated by making rectangular lenses, but image quality, THAT is why lenses are circular.

It's difficult to explain without launching into a complete explanation of quantum electrodynamics, but all of the light that reaches the sensor "goes through" all of the lens, at least in a sense, even if we're just talking about a single photon. A photon doesn't take just one path (unless you make the mistake of trying to figure out which path it took), it takes all possible paths. Weird, but true.

That means that removing glass from a round lens to make a smaller rectangle isn't removing "extra" glass that isn't being used, it would actually be removing glass that is used for imaging (and light collection). By the same token, adding extra glass to make the lens rectangular for purely cosmetic reasons would not only involve a lot of extra expense, that "extra" glass would now also contribute to the imaging probability distribution, so it would need to be as accurately made and as well corrected as the circular lens you are extending. As I've explained here, the larger (faster) you make a lens, the more correction is needed, the more accuracy is required, and the more the price will rise.

Quite apart from that, though, the bokeh (the nature of the out-of-focus areas, particularly the highlights) would look really, really bad.

that is one of the most unrelated answers I've ever read. Disclaimer, I have a PhD in Physics, and I work in the field of AMO physics (O stands for optics). QED has utterly nothing to do with this. Or you can invoke QED for almost any answer about physical phenomena! real life lenses and their engineering are not affected by quantum phenomena. wow.
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blaJul 15 '13 at 7:25

Sure, but that's because everything is quantum at the smallest scale. To me, the interesting question here is "can classical optical theory be used to design modern lenses, or do you have to take into account quantum effects in order to get the performance of modern lenses?" If the answer to that question is "classical optical theory is enough", then QED is a red herring here.
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Philip KendallJul 15 '13 at 11:53

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@anaximander: True, it doesn't mean QED isn't related to the engineering aspects of lens design, but the fact is that it's not; lens-design considerations are entirely encapsulated by the field of optics. This answer is like saying that in order to be a computer programmer, you need to understand quantum mechanics(which describes how transistors work). It's just a hand-wavy attempt to say "it's too difficult to understand," which really means "I don't know."
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BlueRaja - Danny PflughoeftJul 15 '13 at 16:17

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You can't possibly use QED as it is taught (i.e. electro-magnetic interactions modeled through single photon interactions with charged particles) to describe the geometric path of light through glass. The concept you are thinking of is "wave mechanics" -- yes, quantum waves and classical optical waves behave with similar mathematics. But just because "quantum" sounds cooler does not mean that you can use QED to describe ray tracing. Disclaimer: another PhD physicist
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Mark LakataJul 15 '13 at 19:38

This is about the controversial idea that all parts of the image front will collect rays for every pixel.

Diffuse surfaces send ray into every direction, pretty much infinite rays within the small arc that hits the lens. These infinity rays must be directed from a point source to a single pixel. This is hard to do, which is why sharp lenses are hard to find. This is another story.

I shot 3 images wide open and then covered the unused parts with a cut paper rectangle and took 3 more, and saw that the center part was 15% darker when I covered the unused part. the top image is the uncovered one and below is the covered one and as you see the covering is not seen in the frame, it just makes the image 15% darker:

It can be explained in the simplest model of geometric optics. On the object occurs diffuse reflection which may be depicted as several light rays of different brightness in all directions. A larger lens diameter (instead of a smaller rect. shape) may result in a brighter picture.

You are labouring under a delusion. Have you read the answers to the related phy.SE question?
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jwgJul 17 '13 at 6:19

@jwg The diffuse reflection (e.g. lambertian) is correct. Also the conclusion to illuminate all pixels on the detector is right. However PhysicsSE emphasis there is no connection of QED to lens design. I removed QED from the last paragraph.
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Stefan BischofJul 17 '13 at 19:27

I agree that diffuse reflection exists, I just don't agree with the conclusion that you are (both) drawing from it.
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jwgJul 18 '13 at 7:55